Unramified Kummer extensions of prime power degree
type: Item
instance of: scholarly article
title: Unramified Kummer extensions of prime power degree
zbMATH Open document ID: 0705.13004
DOI: 10.1007/BF01165823
published in: Manuscripta Mathematica
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155410
review text: Let \(R\) be a complete local domain with quotient field \(K\) of characteristic 0 and residue field \(k\) of characteristic \(p\), containing \(\zeta\), a primitive \(p^n\)-th root of unity. The author determines the Harrison group \(\mathrm{Gal}(R,G)\) of Galois extensions of \(R\) with group \(G\) cyclic of order \(p^n\). Group \(\mathrm{Gal}(K,G)\cong K^*/K^{*p^n}\) by Kummer theory; also \(\mathrm{Gal}(R,G)\cong \mathrm{Gal}(k,G)\), which is well-known by Artin-Schreier theory. The author's objective is to integrate the two theories by determining \(E_n(R)=\{x\in R^*\mid S_x=R[T]/(T^{p^n}- x)\) is embedded as \(G\)-module and \(R\)-algebra in a \(G\)-Galois extension of \(R\, \}\). Such an integration was achieved by \textit{H. Hasse} [J. Reine Angew. Math. 176, 174--183 (1936; Zbl 0016.05204)] using the Artin-Hasse exponential map. The paper under review obtains a description of \(E_n(R)\) as elements satisfying a certain congruence condition multiplied by values (at elements of \(R\)) of a special set of \(n\) polynomials with coefficients in \(\mathbb Z_p[\zeta]\); the polynomials are defined with the aid of the Artin-Hasse map. The description of \(E_n(R)\) for \(R\) complete extends to general \(R\) by pulling back from the \(p\)-adic completion of \(R\).
reviewed by: Lindsay N. Childs
Mathematics Subject Classification ID: 13B05
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 13J10
Mathematics Subject Classification ID: 13H99
zbMATH DE Number: 4156602
zbMATH Keywords: cyclic Galois group
zbMATH Keywords: Harrison group
author: Cornelius Greither
MaRDI profile type: Publication
cites work: Kummer extensions of rings
cites work: Q5518494
cites work: The Group of Unramified Kummer Extensions of Prime Degree
cites work: Abelsche Galoiserweiterungen von R[X]
cites work: Q3487517
cites work: Q3708099
cites work: Q5518497
cites work: Q5765297
cites work: Die Multiplikationsgruppe der abelschen Körper mit fester Galoisgruppe
cites work: Q3669533
cites work: Patching theorems for projective modules
cites work: Two Fundamental Theorems on Deformations of Polarized Varieties
cites work: Q5682074
cites work: Q5509718
cites work: Q3943047
cites work: A unified Kummer-Artin-Schreier sequence
cites work: Torsors under Abelian p-groups
OpenAlex ID: W2078062135
Recommended article: Galois Endomorphisms of the Torsion Subgroup of One-Parameter Generic Formal groups
Recommended article: Q3116351
Recommended article: Cyclic Galois extensions of commutative rings
Recommended article: Multiplicative Galois Module Structure
Recommended article: Über Galoisgruppen lokaler Körper
Recommended article: Q3760653
Recommended article: On an elementary Abelian \(p\)-extension of a multidimensional local field
Recommended article: On the \(\pi\)-adic theory -- Galois cohomology
Recommended article: On the values of equivariant zeta functions of curves over finite fields
Recommended article: Formal groups and the Galois module structure of local rings of integers.
description: scientific article; zbMATH DE number 4156602